Question

If [2x2 + 1] < 3, then x belongs to (where [ ] denotes the greatest integer function)

Answer 1

Let's start with what exactly the greatest integer function means. It states that if x is a function such that it's greatest integer function is equal to [x], then the value of the function will be less than or equal to the integer.

For example, [2] = 2 (equal to the integer 2)

[2.5] = 2 (less than 2.5, the nearest and smallest integer to 2.5)

Now the question states:

[2x^2 + 1] < 3

Note that, for any negative value of x, the square, i.e., x^2 will remain positive. Thus:

If x = 0 (the least value x can attain)

=> [2x^2 + 1] = [2(0) + 1] = [1] = 1

And 1 < 3, thus x can be equal to 0.

If x = 1, or -1 (both will result in 1 when squared)

=> [2x^2 + 1] = [2(1) + 1] = [3] = 3

But 3 = 3, thus x can't take values of -1, and 1.

However, it can take any values between -1 and 1.

Thus, x belongs to (-1,1)

Note that both the numbers have an open bracket as they both can't be considered as values of x.